F\'{e}lix and Thomas developed string topology of Chas and Sullivan on
simply-connected Gorenstein spaces. In this paper, we prove that the degree
shifted homology of the free loop space of a simply-connected Q-Gorenstein space with rational coefficient is a non-unital and non-counital
Frobenius algebra by solving the up to constant problem. We also investigate
triviality or non-triviality of the loop product and coproduct of particular
Gorenstein spaces.Comment: 27page